Hey guys! Let's dive into a super interesting problem about Dexter, a photographer who's making some serious cash. We're going to break down his earnings and figure out how to represent them using a mathematical inequality. So, grab your thinking caps, and let's get started!
Understanding the Problem: Dexter's Earning Range
The core of this problem lies in understanding the range of Dexter's earnings. We know he makes at least $50, but no more than $100 for each photography session. This "at least" and "no more than" language is super important because it gives us the boundaries we need to work with.
- Minimum Earnings: Dexter never makes less than $50. This means $50 is the lowest amount he can earn.
- Maximum Earnings: On the flip side, Dexter never makes more than $100. So, $100 is the highest amount he can earn.
Our mission is to translate these statements into a mathematical inequality that accurately captures Dexter's earning potential. We'll use the variable e to represent Dexter's earnings for a single session. This e will fall somewhere within the range we just defined.
Decoding the Inequality Options
We've got four options to choose from, and each uses different inequality symbols. Let's break down what these symbols mean so we can pick the right one:
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A. e ≥ 50 or e ≤ 100: This option uses "greater than or equal to" (≥) and "less than or equal to" (≤), but it's connected by the word "or." This means e could be greater than or equal to 50 or less than or equal to 100. Think about it – this would include values like $40 (less than 50) and $110 (more than 100), which doesn't fit our problem because Dexter never earns less than $50 or more than $100.
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B. e > 50 or e < 100: This option is similar to A, but it uses "greater than" (>) and "less than" (<) without the "equal to" part. Again, the "or" is the key issue here. This would mean Dexter's earnings could be any value as long as it's not exactly $50 or $100. It still allows for values outside our range.
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C. 50 < e < 100: This option uses "less than" (<) symbols, suggesting that Dexter's earnings are strictly between $50 and $100. This is close, but it doesn't include the possibility of Dexter earning exactly $50 or exactly $100. Remember, he can earn at least $50 and no more than $100.
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D. 50 ≤ e ≤ 100: This is the winner! This option uses "less than or equal to" (≤) symbols, creating a compound inequality. It reads as "50 is less than or equal to e, and e is less than or equal to 100." This perfectly captures the idea that Dexter's earnings (e) can be any value between $50 and $100, including $50 and $100 themselves.
The Correct Inequality: 50 ≤ e ≤ 100
So, the inequality that best represents Dexter's earnings is D. 50 ≤ e ≤ 100. This inequality tells us that Dexter's earnings (e) will always be within the range of $50 to $100, inclusive. It's a neat and tidy way to express his earning potential for each photography session.
Why This Inequality Works So Well
Let's break down why this inequality is the perfect fit:
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50 ≤ e: This part ensures that Dexter's earnings (e) are always greater than or equal to $50. He never earns less than $50.
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e ≤ 100: This part ensures that Dexter's earnings (e) are always less than or equal to $100. He never earns more than $100.
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Combined: Putting these two parts together, we create a range where e can fall. It's like a sandwich, with $50 and $100 acting as the bread, and Dexter's earnings filling in the middle.
This type of inequality, where a variable is trapped between two values, is called a compound inequality. It's a powerful tool for representing situations where a value has both a minimum and a maximum limit.
Real-World Applications of Inequalities
Understanding inequalities isn't just about solving math problems; it's about understanding the world around us! Inequalities are used in all sorts of real-world situations, such as:
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Budgeting: When you set a budget, you're essentially creating an inequality. For example, if you budget $100 for groceries, your spending (s) must satisfy the inequality s ≤ $100.
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Speed Limits: Speed limits are another example of inequalities in action. If the speed limit is 65 mph, your speed (v) must satisfy the inequality v ≤ 65.
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Age Restrictions: Age restrictions, like needing to be 21 or older to purchase alcohol, are also inequalities. Your age (a) must satisfy the inequality a ≥ 21.
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Grading Scales: Grading scales often use inequalities. For example, a grade of "A" might require a score (s) that satisfies the inequality s ≥ 90.
By understanding inequalities, we can better interpret and navigate these real-world situations. Dexter's photography earnings problem is just a small glimpse into the broader world of inequalities!
Tips for Tackling Inequality Problems
Inequalities can seem a bit tricky at first, but with a few key strategies, you'll be solving them like a pro. Here are some tips to keep in mind:
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Read Carefully: The words used in the problem are super important. Pay close attention to phrases like "at least," "no more than," "greater than," and "less than." These phrases are your clues to choosing the right inequality symbols.
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Identify the Variable: Determine what the variable represents. In Dexter's case, e represents his earnings. Knowing this helps you focus on what you're trying to represent with the inequality.
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Break it Down: For compound inequalities (like the one in our problem), break it down into two separate inequalities. This can make it easier to understand the range the variable must fall within.
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Test Values: Once you've chosen an inequality, test some values to see if they make sense. For example, in Dexter's case, we know his earnings should be between $50 and $100. If we plug in $60, it satisfies the inequality. If we plug in $40 or $110, it doesn't.
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Practice, Practice, Practice: The best way to get comfortable with inequalities is to practice solving them. The more you practice, the easier they'll become!
Common Mistakes to Avoid
When working with inequalities, there are a few common mistakes that students often make. Let's take a look at these so you can steer clear of them:
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Using the Wrong Symbol: The biggest mistake is using the wrong inequality symbol. Remember, "greater than" (>) and "less than" (<) don't include the endpoint, while "greater than or equal to" (≥) and "less than or equal to" (≤) do. Always consider whether the endpoint should be included in the solution.
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Ignoring "Or" vs. "And": When you see the words "or" and "and" in an inequality problem, they have very different meanings. "Or" means either one condition or the other can be true. "And" means both conditions must be true. In our problem, the "or" options were incorrect because Dexter's earnings had to satisfy both conditions (at least $50 and no more than $100).
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Not Reading Carefully: As we mentioned before, careful reading is key. Skimming the problem can lead to overlooking important details, like the "at least" and "no more than" language.
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Not Testing Solutions: Always test your solution by plugging in values. This helps you catch errors and ensure your inequality accurately represents the problem.
Wrapping Up: Inequalities in Action
So, there you have it! We've successfully tackled Dexter's photography earnings problem and learned how to represent his earning range using the inequality 50 ≤ e ≤ 100. We've also explored the real-world applications of inequalities and picked up some handy tips for solving these types of problems. Remember, inequalities are all about setting boundaries and defining ranges, and they're a fundamental tool in mathematics and beyond.
Keep practicing, and you'll become an inequality master in no time! You've got this!